Tools

"... We give two nite axiomatizations of indexed inductive-recursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductive-recursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."

We give two nite axiomatizations of indexed inductive-recursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductive-recursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductive-recursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.

"... We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introductio ..."

We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of&apos;s logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer&apos;s finitely axiomatized theories of inductive-recursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.

"... We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a first-class value in a datatype of descriptions. Moreover, the latter itself has a de ..."

We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a first-class value in a datatype of descriptions. Moreover, the latter itself has a description. Datatype-generic programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently self-supporting setup is achievable without paradox or infinite regress. 1.

"... In a previous paper we have given a representation of continuous functions on streams, both discrete-valued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We ga ..."

In a previous paper we have given a representation of continuous functions on streams, both discrete-valued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We gave also a combinator on the representations of stream processing functions that reflects composition. Streams are the simplest example of a non-trivial final coalgebras, playing in the coalgebraic realm the same role as do the natural numbers in the algebraic realm. Here we extend our previous results to cover the case of final coalgebras for a broad class of functors generalising (×A). The functors we deal with are those that arise from countable signatures of finiteplace untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common prefix, according to the usual definition of the final coalgebra as the limit of a certain inverse chain starting at �. 1

by
Alexandre Buisse, Peter Dybjer
- In International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice, 2007

"... This note is about work in progress on the topic of “internal type theory ” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of de ..."

This note is about work in progress on the topic of “internal type theory ” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of dependent type theory. We discuss how to formalize the notion of category with families inside type theory and how to build initial categories with families. Initial categories with families will be term models which play the role of canonical syntax for dependent type theory. We also discuss the formalization of the result that categories with finite limits give rise to categories with families. This yields a type-theoretic perspective on Curien’s work on “substitution up to isomorphism”. Our formalization is being carried out in the proof assistant Agda 2 developed at Chalmers. 1

"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent ..."

We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-Löf type theory with W-type and one microscopic universe containing only two finite sets in carried out. Then we look at the analysis Martin-Löf theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.

"... Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even ..."

Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even a relation one gets a more general method known as ``partial reflection&apos;&apos;. In this paper we show how one can take advantage of the partiality of the interpretation to uniformly define a family of tactics for equational reasoning that will work in different algebraic structures. The tactics then follow the hierarchy of those algebraic structures in a natural way.

"... GR/S30450/01. 2 A. Setzer, P. Hancock 1 Introduction According to Martin-L&quot;of [19]: &quot;... I do not think that the search for logically ever more satisfac-tory high level programming languages can stop short of anything but ..."

GR/S30450/01. 2 A. Setzer, P. Hancock 1 Introduction According to Martin-L&amp;quot;of [19]: &amp;quot;... I do not think that the search for logically ever more satisfac-tory high level programming languages can stop short of anything but